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| Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| 0nnq | ⊢ ¬ ∅ ∈ Q | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0nelxp 5718 | . 2 ⊢ ¬ ∅ ∈ (N × N) | |
| 2 | df-nq 10953 | . . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))} | |
| 3 | 2 | ssrab3 4081 | . . 3 ⊢ Q ⊆ (N × N) | 
| 4 | 3 | sseli 3978 | . 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N)) | 
| 5 | 1, 4 | mto 197 | 1 ⊢ ¬ ∅ ∈ Q | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2107 ∀wral 3060 ∅c0 4332 class class class wbr 5142 × cxp 5682 ‘cfv 6560 2nd c2nd 8014 Ncnpi 10885 <N clti 10888 ~Q ceq 10892 Qcnq 10893 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-opab 5205 df-xp 5690 df-nq 10953 | 
| This theorem is referenced by: adderpq 10997 mulerpq 10998 addassnq 10999 mulassnq 11000 distrnq 11002 recmulnq 11005 recclnq 11007 ltanq 11012 ltmnq 11013 ltexnq 11016 nsmallnq 11018 ltbtwnnq 11019 ltrnq 11020 prlem934 11074 ltaddpr 11075 ltexprlem2 11078 ltexprlem3 11079 ltexprlem4 11080 ltexprlem6 11082 ltexprlem7 11083 prlem936 11088 reclem2pr 11089 | 
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